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Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

Hayate Kojima, Hiroshi Higashi, Yuichi Tanaka

TL;DR

This work extends Regularization by Denoising (RED) to graph signal denoising, enabling an explicit optimization objective that incorporates a graph denoiser in the regularizer and remains interpretable under mild conditions. It demonstrates that common graph denoisers satisfy RED's requirements and introduces supervised and unsupervised parameter learning via Deep Algorithm Unrolling (DAU) and Noise2Noise, respectively. A graph-filter perspective clarifies how RED can attenuate high-frequency components more effectively than standard Laplacian regularization, improving denoising quality on both synthetic and real-world datasets. Overall, the approach yields a flexible, interpretable, and data-efficient framework that outperforms traditional model-based and data-driven baselines for graph signal denoising, with practical implications for downstream graph tasks.

Abstract

In this paper, we propose an interpretable denoising method for graph signals using regularization by denoising (RED). RED is a technique developed for image restoration that uses an efficient (and sometimes black-box) denoiser in the regularization term of the optimization problem. By using RED, optimization problems can be designed with the explicit use of the denoiser, and the gradient of the regularization term can be easily computed under mild conditions. We adapt RED for denoising of graph signals beyond image processing. We show that many graph signal denoisers, including graph neural networks, theoretically or practically satisfy the conditions for RED. We also study the effectiveness of RED from a graph filter perspective. Furthermore, we propose supervised and unsupervised parameter estimation methods based on deep algorithm unrolling. These methods aim to enhance the algorithm applicability, particularly in the unsupervised setting. Denoising experiments for synthetic and real-world datasets show that our proposed method improves signal denoising accuracy in mean squared error compared to existing graph signal denoising methods.

Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

TL;DR

This work extends Regularization by Denoising (RED) to graph signal denoising, enabling an explicit optimization objective that incorporates a graph denoiser in the regularizer and remains interpretable under mild conditions. It demonstrates that common graph denoisers satisfy RED's requirements and introduces supervised and unsupervised parameter learning via Deep Algorithm Unrolling (DAU) and Noise2Noise, respectively. A graph-filter perspective clarifies how RED can attenuate high-frequency components more effectively than standard Laplacian regularization, improving denoising quality on both synthetic and real-world datasets. Overall, the approach yields a flexible, interpretable, and data-efficient framework that outperforms traditional model-based and data-driven baselines for graph signal denoising, with practical implications for downstream graph tasks.

Abstract

In this paper, we propose an interpretable denoising method for graph signals using regularization by denoising (RED). RED is a technique developed for image restoration that uses an efficient (and sometimes black-box) denoiser in the regularization term of the optimization problem. By using RED, optimization problems can be designed with the explicit use of the denoiser, and the gradient of the regularization term can be easily computed under mild conditions. We adapt RED for denoising of graph signals beyond image processing. We show that many graph signal denoisers, including graph neural networks, theoretically or practically satisfy the conditions for RED. We also study the effectiveness of RED from a graph filter perspective. Furthermore, we propose supervised and unsupervised parameter estimation methods based on deep algorithm unrolling. These methods aim to enhance the algorithm applicability, particularly in the unsupervised setting. Denoising experiments for synthetic and real-world datasets show that our proposed method improves signal denoising accuracy in mean squared error compared to existing graph signal denoising methods.

Paper Structure

This paper contains 23 sections, 2 theorems, 23 equations, 25 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Signal Processing
  4. Regularization by Denoising (RED)
  5. Deep Algorithm Unrolling
  6. Noise2Noise
  7. Graph Signal Denoising
  8. Graph Laplacian Regularization
  9. PnP-ADMM
  10. RED-based Graph Signal Denoising
  11. Are Graph Signal Denoisers Applicable to RED?
  12. LR
  13. GAT and PnP-ADMM
  14. Parameter Learning Using Deep Algorithm Unrolling and Noise2Noise
  15. RED from Graph Filter Perspective
  16. ...and 8 more sections

Key Result

Proposition 1

Suppose that the adjacency matrix $\mathbf{W}$ is constructed with edge weights $W_{i,j}$ for any connected nodes $v_i$ and $v_j$ as: If $\mathbf{W}$ is normalized, the Laplacian regularization denoiser $\mathcal{D}_{\text{LR}}(\cdot)$ defined in eq:lr-closed satisfies $\mathcal{D}_\text{LR}(c\cdot\mathbf{y}) = c \cdot \mathcal{D}_\text{LR}(\mathbf{y})$.

Figures (25)

  • Figure 1:
  • Figure 2:
  • Figure 4: Comparion of $h_\text{lr}(\boldsymbol{\Lambda})$ and $h_\text{red}(\boldsymbol{\Lambda})$ using 3-D point cloud dataset. $\alpha_\text{lr}$ and $\alpha_\text{red}$ are fitted as same as in experiments.
  • Figure 5: Original signal.
  • Figure 6: Observed signal (RMSE:20.03).
  • ...and 20 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof